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Physics Formulas Wiki
. . ---- Please wait while the formulas load :) ---- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement and Units Formulas Estimated uncertainty =\Delta x Percent uncertainty =\frac{\Delta x}{x}\times 100\% Formulas with Summary Uncertainty The estimated uncertainty is the best guess for the uncertainty of a measurement, which may be limited by the precision of the measuring apparatus. Suppose you measure a board with a ruler whose smallest marks are millimeters and find it to be 4.5 cm. A reasonable estimate of the uncertainty would be \pm 0.1 cm so that the answer with uncertainty would be 4.5 \pm 0.1 cm. The percent uncertainty is simply the ratio of uncertainty to the measured value multiplied by 100. For example, in the previous example, the percent uncertainty is \frac{0.1}{4.5}\times 100%\approx 2% here the \approx symbols means "approximately equal to". Significant Figures The number of significant figures is the number of reliably known digits in a number. For example, the number 143.8 has four significant figures while the number 0.0081 has only two significant figures. When multiplying, dividing, adding or subtracting numbers with different numbers of significant figures, the final answer should contain no more significant figures than the number with the least number of significant figures. However, it is best to keep all significant figures throughout the calculation and round the answer at the end. For example, if we measure a rectangle to have base 18.9 cm and height 16.1 cm, the area should be quoted as 304 cm2, not 304.29 cm2. This is because the numbers we started with only had three significant figures so the final answer can have no more than three. Scientific Notation Numbers are commonly expressed in scientific notation. In this convention, there is only one digit to the left of the decimal place. For example, the number 18900 would be 1.89 x 104 and the number .00000037 would be 3.7 x 10-7 Linear Motion Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: x''=position, ''v=velocity, a''=acceleration v=v_0+at\ \ \ _{:bc:} x=x_0+v_0t+\frac{1}{2}at^2\ \ \ _{:bc:} v^2=v_0^2+2a(x-x_0)\ \ \ _{:bc:} \text{If}\ at^2+bt+c=0\ \text{then}\ t=\frac{-b\pm\sqrt{b^2-4ac}}{2a} ('quadratic formula') Formulas with Summary Position and Displacement In one dimension, the '''position' of an object can be specified by one number, x''. The '''displacement' of an object is its change in position. If an object is at position x''1 at time ''t''1 and position ''x''2 at time ''t''2 then its displacement is given by \Delta x = x_2-x_1 The displacement of an object may be different from the '''distance traveled', which is always positive. For example, if a dog walks all the way around a circle of circumference 100 meters its total distance traveled will be 100 meters, but its displacement will be zero since it returned to its starting position. Velocity The average speed refers to how fast an object moved, regardless of direction. It is defined by \text{average speed}=\frac{\text{distance traveled}}{\text{time elapsed}} because distance traveled and time elapsed are always positive numbers, the speed is always positive. In contrast, the average velocity depends on the direction of displacement as well as its magnitude, and can be positive or negative. It is defined by \text{average speed}=\frac{\text{displacement}}{\text{time elapsed}} In symbols this is given by \bar{v}=\frac{x_2-x_1}{t_2-t_1} The instantaneous velocity is the velocity an object has at a particular instant in time. This can be found by calculating the average velocity over an infinitesimal time interval, or equivalently, differentiating the position with respect to time: v=\text{lim}_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}=\frac{dx}{dt} Acceleration The average acceleration is defined as the change in velocity divided by the change in time \text{average acceleration}=\frac{\text{change in velocity}}{\text{time elapsed}} In symbols this is given by \bar{a}=\frac{v_2-v_1}{t_2-t_1} The instantaneous acceleration is defined analogously to the instantaneous velocity as a=\text{lim}_{\Delta t\rightarrow 0}\frac{\Delta v}{\Delta t}=\frac{dv}{dt}=\frac{d^2x}{dt^2} Motion at Constant Acceleration When an object moves with constant acceleration its position and velocity can be simply related to the acceleration and elapsed time. These relationships can be expressed as the familiar equations of one dimensional motion listed in the formulas section above. Suppose that an object is initially at position x''0 and has initial velocity ''v''0 and accelerates with acceleration ''a. After a time t'' the velocity is given by v=v_0+at Its position after a time ''t is given by x=x_0+v_0t+\frac{1}{2}at^2 These formulas can be combined to eliminate the t'' variable in order to yield an expression which relates the velocity to the displacement and acceleration: v^2=v_0^2+2a(x-x_0) Together, these equations can be applied to solve a wide variety of one dimensional motion problems with constant acceleration. Falling Motion A particularly important class of 1-D motion problems study the motion of an object under the influence of gravity. In these problems, the effects of air resistance are often neglected and the magnitude of the acceleration ''a is none other than g''=9.8 m/s2, the acceleration of gravity at the earth's surface. Although ''g is usually defined as a positive number, one must remember that if the coordinate system is oriented so the positive direction is up, then the acceleration of the object is actually negative, that is, a''=-9.8 m/s2. Vectors and Projectile Motion Formulas Key: vi=component of velocity in the ''i direction, ai=component of acceleration in the i'' direction, \theta = angle projectile makes with the horizontal. v_x=v\cos\theta v_y=v\sin\theta \tan\theta=v_y/v_x v=\sqrt{v_x^2+v_y^2} v_x=v_{x0}+a_xt x=x_0+v_{x0}t+\frac{1}{2}a_xt^2 v_x^2=v_{x0}^2+2a_x(x-x_0) v_y=v_{y0}+a_yt y=y_0+v_{y0}t+\frac{1}{2}a_yt^2 v_y^2=v_{y0}^2+2a_y(y-y_0) Formulas with Summary Vectors and Scalars In physics, many objects are '''vectors' which means they have both a magnitude and direction. Examples of vectors are velocity, force and electric field. Objects which only have a magnitude are called scalars. Examples of scalars are mass, temperature and speed. Adding and Subtracting Vectors Vectors can be represented as arrows. Two vectors, V''1 and ''V''2, can be added graphically by moving the end of ''V''2 to the tip of ''V''1 and drawing a new arrow from the end of ''V''1 to the tip of ''V''2 (Fig. 3.1) We can subtract ''V''2 from ''V''1 by adding (-''V''2). We find the negative of a vector simply by reversing the direction of the arrow (Fig 3.2) In addition to representing and adding vectors graphically, we can also represent and add them by components. We represent a vector in component form as a list of numbers in parentheses that specify its (x,y,z) coordinates. In two dimensions we have ''V''1=(v1x,v1y) and ''V''2=(v2x,v2y). Then to add the vectors we simply add their components: \vec{V}_1+\vec{V}_2=(v_{1x}+v_{2x}, v_{1y}+v_{2y}) Vector Decomposition In many physics problems, however, a vector is not given to us in components, but rather, as a magnitude and direction. In this case, we have to use trigonometry to find the components before we can add them. This is particularly common in projectile motion where an object is usually specified to have an initial speed of, say, 20 m/s and an angle of 30° with respect to the horizontal. If | ''V | is the magnitude of the vector and θ is the angle it makes with the horizontal (Fig 3.3), then in two dimensions we have v_x=|V|\cos\theta v_y=|V|\sin\theta We can also find the magnitude and angle of a vector from its components by using the relations V^2=v_x^2+v_y^2 \tan\theta=\frac{v_y}{v_x} Projectile Motion In projectile motion, the linear motion equations are valid for each coordinate separately. In three dimensions, this gives three identical sets of equations: v_x=v_{x0}+a_xt x=x+v_{x0}t+\frac{1}{2}a_xt^2 v_x^2=v_{x0}^2+2a_x(x-x_0) The equations for y and z are obtained by making the substitutions x\rightarrow y or x\rightarrow z respectively. Vectors can also be specified by a magnitude and angles. Let \theta be the angle measured with respect to horizontal. In two dimensions we have: v_x=v\cos\theta v_y=v\sin\theta v=\sqrt{v_x^2+v_y^2} Newton's Laws Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: F'=force, '''F'G=force of gravity, 'F'fric=force of friction, 'F'N=normal force. \sum\mathbf{F}=\mathbf{F_{net}}=m\mathbf{a}\ \ \ _{:bc:} \mathbf{F}_G=m\mathbf{g} \mathbf{F}_{fric}\le\mu \mathbf{F}_N\ \ \ _{:bc:} \mathbf{F}_N=mg\cos\theta \mathbf{F}_{AB}=-\mathbf{F}_{BA} Formulas with Summary Let \mathbf{F}_{fric} = force of friction, \mu = coefficient of kinetic friction and F_N = normal force. When the object is sliding \mu=\mu_k is the coefficient of kinetic friction and when the object is at rest \mu=\mu_s is the coefficient of static friction. Note that we always have \mu_k\le\mu_s which means it takes the same or more force to get an object sliding than it does to keep it sliding. Energy and Work Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: W''=work, ''KE=kinetic energy, PE=potential energy, P''=power, ''E=energy. W=F\Delta r\cos\theta\ \ \ _{:b:} W=\int_C\mathbf{F}\cdot d\mathbf{r}\ \ \ _{:c:} KE=\frac{1}{2}mv^2\ \ \ _{:bc:} PE_{grav}=mgh\ \ \ _{:bc:} PE_{spring}=\frac{1}{2}kx^2\ \ \ _{:b:} P_{avg}=\frac{W}{\Delta t}\ \ \ _{:b:} P=\frac{dW}{dt}\ \ \ _{:c:} P=Fv\cos\theta\ \ \ _{:b:} P=\mathbf{F}\cdot\mathbf{v}\ \ \ _{:c:} E_{initial}=E_{final} Formulas with Summary The work done on an object that is displaced a distance \Delta r by a force F which makes an angle \theta with the direction of motion is given by W=F\Delta r\cos\theta\ \ \ _{:b:} For an object moved along a curve C , this expression can be expressed as a line integral W=\int_C\mathbf{F}\cdot d\mathbf{r}\ \ \ _{:c:} Momentum Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: '''p=momentum, J'=impulse, xCM=position of center of mass. \mathbf{p}=m\mathbf{v}\ \ \ _{:bc:} \mathbf{J}=\mathbf{F}\Delta t=\Delta \mathbf{p}\ \ \ _{:b:} \mathbf{J}=\int\mathbf{F}dt=\Delta \mathbf{p}\ \ \ _{:c:} \mathbf{F}=\frac{d\mathbf{p}}{dt}\ \ \ _{:c:} x_{CM}=(m_ax_a+m_bx_b+...)/(m_a+m_b+...) \sum E_i=\sum E_f and \ \sum p_i=\sum p_f (elastic collision) \sum p_i=\sum p_f (inelastic collision) Formulas with Summary Rotational Motion Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: ''a''c=centripetal acceleration, ''T=period, f''=frequency, θ=angular position, ω=angular velocity, α=angular acceleration. a_c=\frac{v^2}{r}=r\omega^2\ \ \ _{:bc:} T=\frac{1}{f}\ \ \ _{:b:} v=r\omega\ \ \ _{:c:} \omega = \omega_0 +\alpha t\ \ \ _{:c:} \theta=\theta_0+\omega_0t+\frac{1}{2}\alpha t^2\ \ \ _{:c:} \omega=\frac{\Delta \theta}{\Delta t} \alpha =\frac{\Delta \omega}{\Delta t} \omega = 2\pi f \omega^2 = \omega_0^2+2\alpha\Delta\theta Formulas with Summary Torque and Angular Momentum Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: τ=torque, ''I=moment of inertia, 'r'CM=position of center of mass, '''L=angular momentum. \tau = rF\sin\theta\ \ \ _{:b:} \mathbf{\tau}=\mathbf{r}\times\mathbf{F}\ \ \ _{:c:} \sum\mathbf{\tau} =\mathbf{\tau}_{net}=I\mathbf{\alpha}\ \ \ _{:c:} I=\int r^2dm=\sum mr^2\ \ \ _{:c:} \mathbf{r}_{CM}=\sum m\mathbf{r}/\sum m\ \ \ _{:c:} \mathbf{L}=\mathbf{r}\times\mathbf{p}=I\mathbf{\omega}\ \ \ _{:c:} Gravitation Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: F''G=force of gravity, ''U''G=gravitational potential energy. F_G=-\frac{Gm_1m_2}{r^2}\ \ \ _{:b:} U_G=-\frac{Gm_1m_2}{r}\ \ \ _{:b:} Formulas with Summary Static Equilibrium and Elasticity Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: Fs=spring force, ''k=spring constant, x''=displacement, Δ''L=change in length, Δ''V''=change in volume, A''=area, Δ''P=change in pressure, E''=elastic (Young's) modulus, ''B=bulk modulus. Static equilibrium means \sum\mathbf{F}=0 and \sum\mathbf{\tau}=0 F_s=-kx \Delta L=\frac{1}{E}\frac{F}{A}L_0 \Delta V=-\frac{1}{B}V_0\Delta P Formulas with Summary Fluids Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: ρ=density, P''=pressure, ''g=acceleration of gravity, h''=height, ''F''buoy=buoyant force, ''A=area, v''=velocity, ''y=vertical position. \rho=m/V\ \ \ _{:b:} P=F/A\ \ \ _{:b:} P=P_0+\rho g h\ \ \ _{:b:} P_{out}=P_{in}\ \ \ _{:b:} F_{buoy}=\rho_{liquid}V_{object}g\ \ \ _{:b:} A_1v_1=A_2v_2\ \ \ _{:b:} P+\rho g y+\frac{1}{2}\rho v^2=constant\ \ \ _{:b:} v_1=\sqrt{2g(y_2-y_1)} Vibrations and Waves Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: F''s=spring force, ''T''s/p=spring/pendulum period, ''T=period, f''=frequency, ''v=velocity of wave, λ=wavelength, E''SHO=energy of simple harmonic oscillator, ''F''pend=restoring force of pendulum. F_s=-kx\ \ \ _{:b:} T_s=2\pi\sqrt{m/k}\ \ \ _{:b:} T_p=2\pi\sqrt{l/g}\ \ \ _{:b:} T=1/f\ \ \ _{:b:} v=f\lambda\ \ \ _{:b:} E_{SHO}=\frac{1}{2}mv^2+\frac{1}{2}kx^2 x(t)=A\cos(\omega t+\theta_0) v(t)=-\omega A\sin(\omega t+\theta_0) a(t)=-\omega^2A\cos(\omega t+\theta_0) F_{pend}=-mg\sin\theta\approx -mg\theta Sound Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: ''v=velocity of sound, β=sound level, λ=wavelength, f'=doppler shifted frequency. v=f\lambda\ \ \ _{:b:} \beta \text{(in dB)} =10\log \frac{I}{I_0} f_{beats}=f_1-f_2 \lambda_1=2L (open tube) f_n=nf_1 (open tube) \lambda_1=4L (closed at one end) f_n=(2n+1)f_1 (closed at one end) f'=f/(1-\frac{v_{source}}{v_{sound}}) (source moving toward observer) f'=f/(1+\frac{v_{source}}{v_{sound}}) (source moving away from observer) Formulas with Summary Temperature and Thermal Expansion Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: T''=temperature, Δ''L=change in length, α=coefficient of linear expansion, Δ''V''=change in volume, β=coefficient of volume expansion, n''=number of mols, ''R=universal gas constant, k''B=Boltzmann's constant, ''KE''avg=average kinetic energy of gas molecules at temperature ''T, v''rms=root mean square velocity. T(K)=T(C^{\circ})+273.15 \Delta L=\alpha L_0\Delta T\ \ \ _{:b:} \Delta V=\beta V_0\Delta T PV=nRT=Nk_BT\ \ \ _{:b:} KE_{avg}=\frac{3}{2}k_BT (ideal gas) \ \ _{:b:} v_{rms}=\sqrt{3RT/M}=\sqrt{3k_BT/\mu}\ \ \ _{:b:} W=-P\Delta V (work done on a system) \ \ _{:b:} Heat Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: ''U=internal energy, Q''=heat, ''m=mass, c''=specific heat, ''L''f=heat of fusion, ''L''V=heat of vaporization, ''H=rate of heat transfer, k'' thermal conductivity, ''A=area, L''=length, ''I=intensity of radiation emitted by blackbody, e''=emissivity, σ=Stefan-Boltzmann constant. U=\frac{3}{2}nRT Q=mc\Delta T Q=mL_f (melting/freezing) Q=mL_V (vaporizing/condensing) H=(kA\Delta T)/L (Heat transfer) \ \ \ _{:b:} I=e\sigma AT^4 Thermodynamics Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: Δ''U=change in internal energy, e''=efficiency, ''T''H=temperature of hot reservoir, ''T''C=temperature of cold reservoir. \Delta U=Q+W\ \ \ _{:b:} W=-P\Delta V\ \ \ _{:b:} (work done ''on a system) e=|W|/|Q_H|\ \ \ _{:b:} e=(T_H-T_C)/T_H\ \ \ _{:b:} \Delta Q=T\Delta S (T is constant) Electric Charge Forces and Fields Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: q''=charge, ''r =separation, k''=Coulomb's law constant, ε0=vacuum permittivity, ''Q=total charge enclosed. F=\frac{kq_1q_2}{r^2}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}\ \ \ _{:bc:} \mathbf{E}=\frac{\mathbf{F}}{q}\ \ \ _{:bc:} \oint \mathbf{E}\cdot d\mathbf{A}=Q/\epsilon_0\ \ \ _{:c:} Electric Potential and Capacitance Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: U''E=electric potential energy, ''V=voltage, E''=electric field, ''C=Capacitance, Q''=charge on capacitor, ''A=area of capacitor plate, d''=separation of plates, ''U''C=energy stored in capacitor. U_E=qV=\frac{kq_1q_2}{r}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}\ \ \ _{:bc:} E_{avg}=-V/d\ \ \ _{:b:} E=-\frac{dV}{dr}\ \ \ _{:c:} V=k(\frac{q_1}{r_1}+\frac{q_2}{r_2}+\frac{q_3}{r_3}+...)\ \ \ _{:b:} V=\frac{1}{4\pi\epsilon_0}\sum\frac{q_i}{r_i}\ \ \ _{:c:} C=Q/V\ \ \ _{:bc:} C=\epsilon_0A/d\ \ \ _{:bc:} U_C=\frac{1}{2}QV=\frac{1}{2}CV^2\ \ \ _{:bc:} Currents and Resistance Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: ''I=current, R''=resistance, ''P=power, ρ=resistivity, L''=length, ''A=area, N''=number of charge carriers per unit volume, ''v''d=drift velocity, '''J'=current density. I_{avg}=\frac{\Delta Q}{\Delta t}\ \ \ _{:b:} I=\frac{dQ}{dt}\ \ \ _{:c:} V=IR\ \ \ _{:bc:} P=IV\ \ \ _{:bc:} R=\rho L/A\ \ \ _{:bc:} I=Nev_dA\ \ \ _{:c:} \mathbf{E}=\rho\mathbf{J}\ \ \ _{:c:} Circuits Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: R''=resistance, ''C=capacitance, V''c=voltage on capacitor, ''Q''c=charge on capacitor, τ=time constant. R_{series}=R_1+R_2+R_3+...\ \ \ _{:b:} R_{series}=\sum_iR_i\ \ \ _{:c:} 1/R_{parallel}=1/R_1+1/R_2+1/R_3+...\ \ \ _{:b:} 1/R_{parallel}=\sum_i 1/R_i\ \ \ _{:c:} C_{parallel}=C_1+C_2+C_3+...\ \ \ _{:b:} C_{parallel}=\sum_iC_i\ \ \ _{:c:} 1/C_{series}=1/C_1+1/C_2+1/C_3+...\ \ \ _{:b:} 1/C_{series}=\sum_i 1/C_i\ \ \ _{:c:} V_C=V(1-e^{-t/RC}) (charging) Q_C=Q_0(1-e^{-t/RC}) (charging) V_C=V_0e^{-t/RC} (discharging) Q_C=Q_0e^{-t/RC} (discharging) \tau=RC (time constant) Magnetism Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: ''q=charge, F''B=magnetic force, ''v=velocity, B''=magnetic field, θ=angle, ''L=length, μ0=vacuum permeability, r'' =separation, ''n=number of loops of wire per unit length, l''=length of one loop of wire, I=current, τ=torque on current loop, ''N=number of loops, A''=area. F_B=qvB\sin\theta\ \ \ _{:b:} \mathbf{F}_B=q\mathbf{v}\times\mathbf{B}\ \ \ _{:c:} F_B=BIL\sin\theta\ \ \ _{:b:} \mathbf{F}=\int Id\mathbf{l}\times\mathbf{B}\ \ \ _{:c:} \oint\mathbf{B}\cdot d\mathbf{l}=\mu_0 I\ \ \ _{:c:} B=\frac{\mu_0I}{2\pi r}\ \ \ _{:b:} d\mathbf{B}=\frac{\mu_0}{4\pi}\frac{Id\mathbf{l}\times\mathbf{r}}{r^3}\ \ \ _{:c:} B_s=\mu_0nI/l\ \ \ _{:c:} \tau=NIAB\sin\theta Electromagnetic Induction Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: φm=magnetic flux, ε=EMF, ''v=velocity, V''P/S=voltage on primary/secondary coil, ''N''P/S=turns of wire in primary/secondary coil, ''U''L=energy stored in an inductor, ''L=inductance. \phi_m=BA\cos\theta\ \ \ _{:b:} \phi_m=\int\mathbf{B}\cdot d\mathbf{A}\ \ \ _{:c:} \mathcal{E}_{avg}=-\frac{\Delta\phi_m}{\Delta t}\ \ \ _{:b:} \mathcal{E}=\oint\mathbf{E}\cdot d\mathbf{l}=-\frac{d\phi_m}{dt}\ \ \ _{:c:} \mathcal{E}=Blv\ \ \ _{:b:} \frac{V_S}{V_P}=\frac{N_S}{N_P} (Transformer equation) \mathcal{E}=-L\frac{dI}{dt}\ \ \ _{:c:} U_L=\frac{1}{2}LI^2\ \ \ _{:c:} Electromagnetic Waves Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: B'''=magnetic field, '''J=current density, E=electric field, U''EM=energy stored in electromagnetic field, \bar{I} =average intensity, ''c=speed of light, P''=radiation pressure. \nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t} v=f\lambda\ \ \ _{:b:} U_{EM}=\frac{1}{2}\epsilon_0E^2+\frac{1}{2\mu_0}B^2=\epsilon_0E^2 \bar{I}=\frac{1}{2}\epsilon_0cE_0^2 P=\frac{\bar{I}}{c} (total absorption) P=\frac{2\bar{I}}{c} (total reflection) Reflection and Refraction Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: θ=angle, ''f=focal length, R''=radius of curvature, ''d''i/o=distance to image/object, ''h''i/o=height of image/object, ''m=magnification, n''=index of refraction, θC=critical angle. \theta_{incident}=\theta_{reflected} f=\frac{R}{2}\ \ \ _{:b:} \frac{1}{d_i}+\frac{1}{d_o}=\frac{1}{f}\ \ \ _{:b:} m=\frac{h_i}{h_o}=-\frac{d_i}{d_o}\ \ \ _{:b:} n=\frac{c}{v}\ \ \ _{:b:} n_1\sin\theta_1=n_2\sin\theta_2\ \ \ _{:b:} \sin\theta_C=\frac{n_2}{n_1}\ \ \ _{:b:} Interference and Diffraction Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: ''d=distance between slits, λ=wavelength, m''=an integer, ''D=width of single slit, I''=intensity of light, θp=polarizing angle. d\sin\theta = m\lambda (constructive) \ \ \ _{:b:} d\sin\theta=(m+\frac{1}{2})\lambda (destructive) D\sin\theta=m\lambda,\ m\ne0 (single slit minima) d \sin\theta=m\lambda (diffraction grating maxima) I_{out}=I_{in}\cos^2\theta (polarization) \tan\theta_p=n_2/n_1 Optical Instruments Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: ''M=magnifying power, f''=focal length, ''N=eye's near point (usually use N=25 cm), f''o/e focal length of objective/eyepiece, θm=minimum resolvable angular separation, ''D=diameter of telescope. M=\frac{N}{f} (magnifying glass; eye focused at infinity) M=\frac{N}{f}+1 (magnifying glass; eye focuses at near point N) M=-\frac{f_o}{f_e} (telescope) \theta_m=1.22\lambda/D Special Relativity Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: γ=relativistic factor, Δ''t''0=separation of time in rest frame, L''0=length in rest frame, ''p=momentum, m''0=rest mass, ''c=speed of light, E''=energy, ''v''3= the sum of velocities ''v''1 and ''v''2 when added relativistically. \gamma=\frac{1}{\sqrt{1-v^2/c^2}} \Delta t=\gamma\Delta t_0 L=L_0/\gamma p=\gamma m_0 v E_0=m_0c^2 E=\gamma m_0c^2 E^2=p^2c^2+m_0^2c^4 v_3=\frac{v_1+v_2}{1+v_1v_2/c^2} Quantum Theory and the Atom Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: ''E=energy, h''=Planck's constant, ''f=frequency, KE=kinetic energy of ejected electron, φ=work function, p''=momentum, θ=Compton scattering angle, ''R=Rydberg constant, n''=an integer, ''Z=atomic number, Δ''x''=uncertainty in position. E=hf\ \ \ _{:b:} KE=hf-\phi (photoelectric effect) \ \ \ _{:b:} \lambda = h/p\ \ \ _{:b:} \lambda'=\lambda+\frac{h}{m_0c}(1-\cos\theta) \frac{1}{\lambda}=R(\frac{1}{2^2}-\frac{1}{n^2}),\ n\ge3 (Balmer series of H) \frac{1}{\lambda}=R(\frac{1}{1^2}-\frac{1}{n^2}),\ n\ge2 (Lyman series of H) \frac{1}{\lambda}=R(\frac{1}{3^2}-\frac{1}{n^2}),\ n\ge4 (Paschen series of H) E_n=-(13.6 eV)\frac{Z^2}{n^2} h/(2\pi)\le\Delta x\Delta p h/(2\pi)\le\Delta E\Delta t Nuclear Physics and Radioactivity Formulas Note: The _{:b:}\ ,\ _{:c:} and _{:bc:} symbols indicate that a formula is on the AP Physics B, C or both B and C formula sheets. Key: E''=energy, ''m=mass, N=number of radioactive nuclei, λ=decay constant, ''T''1/2=half-life. E= mc^2\ \ \ _{:b:} \alpha particle = 2 neutrons + 2 protons \beta -decay: neutron \rightarrow proton+electron+neutrino N=N_0e^{-\lambda t} T_{1/2}=(\ln 2)/\lambda =References= Fundamental Constants Note: This material is included in the AP physics B and C equation sheets. Unit Symbols and Definitions Some of this information is on the AP physics B and C equation sheets. Geometry and Trigonometry This material can be found on both the AP physics B and C equation sheets. Key: A =area, C =circumference, V =volume, S =surface area, b =base, h =height, l =length, w =width, r =radius. Rectangle A=bh Triangle A=\frac{1}{2}bh Circle A=\pi r^2 C=2\pi r Rectangular Solid V=lwh Cylinder V=\pi r^2l S=2\pi rl+2\pi r^2 Sphere V=\frac{4}{3}\pi r^3 S=4\pi r^2 Right Triangle a^2+b^2=c^2 \sin\theta=a/c \cos\theta = b/c \tan\theta = a/b Calculus This material is found on the AP physics C equation sheet only. \frac{df}{dx}=\frac{df}{du}\frac{du}{dx} \frac{d}{dx}(x^n)=nx^{n-1} \frac{d}{dx}(e^{a\cdot x})=ae^{a\cdot x} \frac{d}{dx}(\ln x)=\frac{1}{x} \frac{d}{dx}(\sin x)=\cos x \frac{d}{dx}(\cos x)=-\sin x \int x^n\ dx=\frac{1}{n+1} x^{n+1},\ n\ne -1 \int e^{a\cdot x}\ dx=\frac{1}{a}e^x \int\frac{dx}{x}=ln|x| \int\cos x\ dx=\sin x \int\sin x\ dx=-\cos x Prefixes Note: This material is included in the AP physics B and C equation sheets. Unit Conversions Length 1 in = 2.54 cm 1 cm = 0.3937 in 1 ft = 30.48 cm 1 m = 39.37 in = 3.281 ft 1 mi = 5280 ft = 1.609 km 1 fermi = 1 femtometer(fm)=10-15 m 1 Angstrom = 10-10 m 1 light-year (ly)=9.461 x 1015 m 1 parsec = 3.26 ly=3.06 x 1016 m Volume 1 liter(L)=1000 mL =1000 cm3=1.0 x 10-3 m3 1 gal(U.S) = 4qt (U.S) = 3.785 L 1 quart (U.S.)= 2 pints (U.S.)=586 mL 1 pint (British) = 1.20 pints (U.S.) = 586 mL 1 m3=35.31 ft3 Speed 1 mi/h = 1.467 ft/s = 1.609 km/h = .447 m/s 1 km/h = .278 m/s = .621 mi/h 1 ft/s = 0.305 m/s = 0.682 mi/h 1 m/s = 3.281 ft/s = 3.600 km/h = 2.237 mi/h 1 knot = 1.151 mi/h = 0.5144 m/s Angle 1 radian (rad) =57.30° 1 ° =0.01745 rad 1 rev/min (rpm)= 0.1047 rad/s Time 1 day = 8.64 x 104 s 1 year =3.156 x 107 s Mass 1 atomic mass unit (u) =1.6605 x 10-27 kg 1 kg = 0.0685 slug kg has a weight of 2.20 lb where g = 9.80 m/s2 Force 1 lb = 4.45 N 1 N = 105 dyne = 0.225 lb Energy and Work 1 J = 107 ergs = 0.738 ft \cdot lb 1 kcal =4.186 x 103 J = 3.97 BTU 1 eV =1.602 x 10-19 J 1 kWh =3.60 x 106 J Power 1 W = 1J/s=0.738 ft \cdot lb/s = 3.42 BTU 1 hp =550 ft \cdot lb/s=746 W Pressure 1 atm = 1.013 bar = 1.013 x 105 N/m2 =760 torr 1 Pa = 1 N/m2 For more help with physics check out the app Physics X on our website Knowvio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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